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Mirrors > Home > MPE Home > Th. List > Mathboxes > ineccnvmo | Structured version Visualization version GIF version |
Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.) |
Ref | Expression |
---|---|
ineccnvmo | ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5644 | . . 3 ⊢ Rel ◡𝐹 | |
2 | id 22 | . . . 4 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧) | |
3 | 2 | inecmo 34462 | . . 3 ⊢ (Rel ◡𝐹 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥) |
5 | brcnvg 5441 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)) | |
6 | 5 | el2v 34329 | . . . 4 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦) |
7 | 6 | rmobii 3282 | . . 3 ⊢ (∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
8 | 7 | albii 1895 | . 2 ⊢ (∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
9 | 4, 8 | bitri 264 | 1 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 834 ∀wal 1629 = wceq 1631 ∀wral 3061 ∃*wrmo 3064 Vcvv 3351 ∩ cin 3722 ∅c0 4063 class class class wbr 4786 ◡ccnv 5248 Rel wrel 5254 [cec 7894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ec 7898 |
This theorem is referenced by: ineccnvmo2 34467 |
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